Problem: Solve for $x$ : $ 8|x + 8| - 9 = -2|x + 8| + 10 $
Answer: Add $ {2|x + 8|} $ to both sides: $ \begin{eqnarray} 8|x + 8| - 9 &=& -2|x + 8| + 10 \\ \\ { + 2|x + 8|} && { + 2|x + 8|} \\ \\ 10|x + 8| - 9 &=& 10 \end{eqnarray} $ Add ${9}$ to both sides: $ \begin{eqnarray} 10|x + 8| - 9 &=& 10 \\ \\ { + 9} &=& { + 9} \\ \\ 10|x + 8| &=& 19 \end{eqnarray} $ Divide both sides by ${10}$ $ \dfrac{10|x + 8|} {{10}} = \dfrac{19} {{10}} $ Simplify: $ |x + 8| = \dfrac{19}{10}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 8 = -\dfrac{19}{10} $ or $ x + 8 = \dfrac{19}{10} $ Solve for the solution where $x + 8$ is negative: $ x + 8 = -\dfrac{19}{10} $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& -\dfrac{19}{10} \\ \\ {- 8} && {- 8} \\ \\ x &=& -\dfrac{19}{10} - 8 \end{eqnarray} $ Change the ${ - 8}$ to an equivalent fraction with a denominator of $10$ $ x = - \dfrac{19}{10} {- \dfrac{80}{10}} $ $ x = -\dfrac{99}{10} $ Then calculate the solution where $x + 8$ is positive: $ x + 8 = \dfrac{19}{10} $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& \dfrac{19}{10} \\ \\ {- 8} && {- 8} \\ \\ x &=& \dfrac{19}{10} - 8 \end{eqnarray} $ Change the ${ - 8}$ to an equivalent fraction with a denominator of $10$ $ x = \dfrac{19}{10} {- \dfrac{80}{10}} $ $ x = -\dfrac{61}{10} $ Thus, the correct answer is $x = -\dfrac{99}{10} $ or $x = -\dfrac{61}{10} $.